ANÁLISIS COMPARATIVO

Chapter 4

MEASUREMENT OF THE ZEEMAN SPLITTING OF THE SODIUM 3S and 3P ATOMIC STATES BY

SATURATED ABSORPTION SPECTROSCOPY

Magnetic field induced second harmonic generation (SHG) [14] and sum frequency mixing (SFM) (Chapter 5) in sodium vapour use the sodium 3P atomic states as nonresonant intermediate states for two-photon absorption. The relatively small laser detuning from the 3S-3P dipole transition provides some single photon resonant enhancement for tlie nonlinear

optical processes. In these cases, the laser detuning is always much greater than the Doppler width or the magnetic splitting of the 3S-3P D line transitions and the perturbing effect of the transverse magnetic field on tlie energy of the sodium 3P states can be neglected. However, when the SFM nonlinear process is "two-step" with simultaneous single and two-photon resonant enhancement (Chapter 6), then the magnetic splittings of the sodium D line transitions become relevant and the laser detuning from each individual hyperfine transition frequency is

required.

It was desirable to have an in situ experimental method of measuring the frequency splittings of the various D line hypeifrne transitions with an applied magnetic field. These could then be compared with the theoretical Zeeman splittings to calibrate the magnetic field and obtain laser detuning measurements for the SFM nonlinear optical process. The Zeeman energy splittings of the sodium 3Si/2 and 4D atomic states can be measured by Doppler-free two-photon absorption [89,90]. A relatively simple experimental technique of obtaining high- resolution. Doppler-free spectra of the sodium 3S-3P single photon transitions is to use saturated absorption spectroscopy.

A few investigations have previously been carried out to examine the Zeeman splitting of the sodium D lines with a Doppler-free laser spectroscopic method. The magnetic splitting and line intensities of the 3Si/2-3Pi/2 (^i) [91] and 3Sj/2-3P3/2 (D2) [92] sodium transitions

have been studied by Windliolz and Musso. Tliey used laser induced fluorescence in an atomic beam to obtain Doppler-free spectra up to magnetic field strengths of ~ 300 G. The line intensity of a given D line hyperfine transition varied with magnetic field strength because of

the change in coupling of the nuclear and electronic angular momenta. The experimental line intensities and frequency splittings agreed well with their developed theoretical calculations. Polarisation spectroscopy has also been used to study Zeeman splittings of the sodium line [93]. A heterodyning technique produced high resolution experimental spectra which agreed

favourably with theory. However, interpretation of the Zeeman spectrum was not straightforward due to the many additional cross-over resonances and comparison between theory and experiment would have been impossible without prior knowledge of the hyperfine structure and g-factors of the atomic states.

Although, as shown later in this chapter, the use of saturated absorption (or saturation) spectroscopy in determining Zeeman structure is also limited by cross-over resonances, it is believed that no previous study has used this laser spectroscopic technique for measuring magnetic field sphttings of the sodium Dj and D2 resonance lines.

4 .1 SATURATED, ABSORPTION SPECTHDJSnOEY 4.1.1 Basic Principles of Saturation Spectroscopy

The resonant interaction of a strong light field with an inhomogeneously broadened

atomic transition can produce a velocity selective modification to the level populations in the ground and excited state. Consider the two level atom shown in Figure 4.1 which is interacting with a strong narrow linewidth laser field at frequency co.

When the detuning A of the laser field from the atomic transition is less than the Doppler width Aa)o then the effect of the strong field is to selectively excite a homogenous velocity group from the ground state. The Doppler width of the transition is

, 2k_T

AUj^(HWHM) = ^ / — j(n2 (4.1.1)

Me

where M is the mass of the atom, kg is Boltzmann’s constant and T is the absolute temperature

v=

V

Figure 4.1 : Simple two level atom with resonant frequency interacting with an intense light field and the

inhomogeneous level populations as functions of atomic velocity (from [95]).

distributions by the intense pump laser field is probed by measuring the absorption of a weak counter- propagating laser beam of the same fiequency as the pump beam. The intensity of the

probe beam is weak so that it does not also perturb the atomic populations. Both waves are usually obtained from the same laser source.

When the laser frequency is scanned across the inhomogenous atomic transition, the

pump and probe beams will interact with equal magnitude velocity groups but opposite in

direction due to the beams being counter-propagating. Both waves will only interact with the

same homogeneous velocity ^oup when tlie zero velocity component is excited. The probe wave then experiences less absorption from the ground state due to the hole burning effect of the pump field and the decrease in probe absoiption can be detected. This is usually done by modulating the pump beam and detecting the probe absorption change at the modulation

frequency with a phase-sensitive detector.

The change (or saturation) in the absorption of the probe wave occurs at the zero velocity frequency of the atomic transition, ie. the natural atomic frequency, and over a small

frequency range due to the narrow homogeneous width of the burnt hole by the pump field. This is the basic principle of saturation spectroscopy to obtain high resolution Doppler-free

In the absence of the saturating pump field, the absorption of the probe wave cOp in the inhomogeneously broadened two level atomic medium is given by

-Mv^/2kgT

/ M

[r f G

( 5 _ k v f + f

where <Jq is the peak atomic absorption cross-section, ANq is the thermal population difference

between the ground and excited state, Ô = 0)p - Cû2i, Y is the inverse of the excited state decay time and the normalised Maxwell distribution of atomic velocities has been used. The line shape of Kg (cOp) is the convolution of a Lorentzian with a Gaussian lineshape and is named a Voigt profile. With a resonant saturating field at frequency o)g, the absorption coefficient for the probe wave becomes

o f

1-

(co^i -cOg- k^v)^ + [y (1 +

]

exp ( - Mv^ / 2kgT) dv . (4.1.3) (6-kpV)

This is again a Voigt profile but now a "hole" has been created in the absorption of the probe wave at the frequency of the pump wave. The power broadened width (HWHM) of the burnt

hole is Y (1+ G)& where G is the saturation parameter

G = Ig/Isat" (4.1.4)

Igat is the saturation intensity (Wm-^) of the atomic transition and is defined as the pump laser intensity required to make the probe absorption coefficient one half of the unsaturated value «q.

For a simple two-level atomic transition and only spontaneous decay from the excited state

Several factors can contribute to the measured homogeneous linewidth of the saturated

absoiption spectrum [148]. The relative magnitudes of these contributions is important when using this nonlinear spectroscopic technique to study closely spaced transition frequencies which are expected for the Zeeman structure of the sodium 3S-3P resonances. If the experimental linewidth is large compared to the magnetic splitting then the different Zeeman

hyperfine transitions may not be resolved.

(a) Natural broadening - If spontaneous emission from the excited state is the only

decay process in the two level atomic system, then the measured saturated absorption linewidth Y (HWHM) is equal to the inverse of the lifetime % of the excited state.

Am (HWHM) = Y = 1/T (4.1.6)

For sodium, the lifetime of the 3P states is 16ns and tliis gives a natural linewidth F (= y/2tc) =

10 MHz (HWHM).

(b) Power broadening - The linewidth of the burnt hole in the ground state population

distribution depends upon tire intensity of the pump field as shown in equation (4.1.3). This is

due to induced stimulated emission and absorption by the pump field decreasing the effective

lifteime of the excited state. The increase in linewidth is (1+G)& times the homogeneous linewidth due to natural decay and collisions. (See also § 6.5.1)

Aw = Aco^ (1+0* (4.1.7)

The saturation intensity for sodium 3S-3P transitions is ~6.4 mW cm-^ therefore the pump beam intensity should be comparable to or a few times this value to avoid serious power broadening.

level atomic system is approximated to be infinite, there is a finite interaction time of an atom with the pump and probe fields due to the thermal motion of the atom through the laser beam. This interaction time T inti'oduces transit-time broadening with a linewidth contribution (HWHM) of '-2.79/T. The motion of an atom with velocity v through a laser beam of diameter

d gives the broadening as

Am (HWHM) = 2.79 ^ (4.1.8)

An average thennal velocity of --500 ms'^ and a laser beam diameter of -1 mm gives a broadening of -220 kHz.

(d) Spherical wavefronts - The wavefront curvature of a Gaussian laser beam restricts the effective transit time and can broaden the saturated absorption linewidth.

The linewidth contiibution is

Am (HWHM) = n /21n2‘ vw/RX, (4.1.9)

where v is the velocity of the atom through the laser beam of width w, wavefront radius of curvature R and wavelength X, In order that the curvature of the wave surface does not much broaden the homogeneous linewidth R » IIw % , eg. for w = 1 mm, X = 600 nm => R » 5 m. In general, a very flat wave suiface should be used for very high resolution spectroscopy but with the above values and v -500 ms“^ the broadening is only -100 kHz. (e) Residual Doppler broadening - If the counter-propagating pump and probe beams are not collinear but cross at a small angle 0, there is a residual Doppler broadening due to slightly different velocity groups being probed and pumped. The linewidth contribution is

Aco (HWHM) « 0.6 0 Acod (4.1.10)

where AcO£>(HWHM) is the Doppler width given in equation (4.1.1). The sodium 3S-3P Doppler width (FWHM), at normal sodium vapour temperatures, is -1.5 GHz, which

processes (see also § 6.4). This gives an additional increase in the homogeneous linewidth of

Ad (HWHM) = p P . (4.1.11)

where P (MHz/mbar) is the pressure broadening coefficient and P (mbar) is the perturber gas pressure. Typical values of P ~25 MHz/mbar but if the buffer gas pressure is low then